We propose a threshold-type algorithm to the $L^2$-gradient flow of the Canham-Helfrich functional generalized to $\mathbb{R}^N$. The algorithm to the Willmore flow is derived as a special case in $\mathbb{R}^2$ or $\mathbb{R}^3$. This algorithm is constructed based on an asymptotic expansion of the solution to the initial value problem for a fourth order linear parabolic partial differential equation whose initial data is the indicator function on the compact set $\Omega_0$. The crucial points are to prove that the boundary $\partial\Omega_1$ of the new set $\Omega_1$ generated by our algorithm is included in $O(t)$-neighborhood from $\partial\Omega_0$ for small time $t>0$ and to show that the derivative of the threshold function in the normal direction for $\partial\Omega_0$ is far from zero in the small time interval. Finally, numerical examples of planar curves governed by the Willmore flow are provided by using our threshold-type algorithm.

翻译：本文提出一种阈值型算法，用于求解推广至$\mathbb{R}^N$的Canham-Helfrich泛函的$L^2$-梯度流。作为特例，该算法可推导出$\mathbb{R}^2$或$\mathbb{R}^3$中Willmore流的求解方法。该算法基于四阶线性抛物型偏微分方程初值问题解的渐近展开构建，其中初始数据为紧集$\Omega_0$上的示性函数。关键点在于：证明由算法生成的新集合$\Omega_1$的边界$\partial\Omega_1$在小时步$t>0$条件下包含于$\partial\Omega_0$的$O(t)$邻域内，并证明显式上述解在$\partial\Omega_0$法向导数在小时时间区间内非零。最后，利用该阈值型算法提供Willmore流驱动的平面曲线数值算例。